Answer
$\displaystyle \frac{5}{2}+\ln 3-\ln 2$
Work Step by Step
FTC: Let $f$ be a continuous function defined on the interval $[a, b]$,
and let $F$ be any antiderivative of f defined on $[a, b]$. Then
$\displaystyle \int_{a}^{b}f(x)dx=\left[F(x)\right]_{a}^{b}=F(b)-F(a)$.
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$f(x)=x+\displaystyle \frac{1}{x}$, which is continuous on [$2,3$].
First, find an antiderivative $F(x)$:
$\displaystyle \int f(x)dx=\frac{x^{2}}{2}+\ln|x|+C$
(we take C=0, since by FTC we need ANY antiderivative)
$F(x)=\displaystyle \frac{x^{2}}{2}+\ln|x|$
Next, evaluate:
$F(3)=\displaystyle \frac{9}{2}+\ln 3$
$F(2)=\displaystyle \frac{4}{2}+\ln 2=2+\ln 2$
Finally, apply the theorem:
$\displaystyle \int_{2}^{3}f(x)dx=F(3)-F(2)$
$=\displaystyle \frac{9}{2}+\ln 3-(2+\ln 2) $
$=\displaystyle \frac{5}{2}+\ln 3-\ln 2\approx$2.90546510811
